Engineering analysis + design software

User Area > Advice

Natural Coordinate System

In order to make the finite element method as generally applicable as possible, a natural coordinate system is required. Typically it is specified in terms of (x) for line elements (bars, beams), (x, h) for surface elements (plates, plane membranes) and (x, h, z) for volume elements (solids).

The coordinate axis (x,h,z) are curvilinear coordinates and can be thought of a representing the more usual Cartesian system (x,y,z) locally within the element. The natural coordinates are chosen such that each coordinate axis varies between –1 and +1 irrespective of the actual dimensions of the element – in essence this is a normalised coordinate system.

This natural coordinate system permits the use of numerical integration and allows element integrals to be performed over simple and predetermined limits rather than over the limits determined during the solution as a result of element deformation and geometry change. The use of numerical integration immediately renders a tremendous flexibility to the finite element method – making it applicable to many different classes of problem in all scientific disciplines. The natural coordinate system also facilitates the use of element shape functions and their associated benefits, together with a vast range of elements that may be formed easily from a standard set of basic equations, rather than a new basis for each element.

 A typical one-dimensional natural coordinate system for a two-noded line element is represented as follows

And for a three-noded line element as

The coordinate system is “fixed” to the element and deforms with it such that, for example, the x coordinate for a bar will, at all points, be along the length of the bar. The origin of the system is at the centre of the element (x=0).

For two dimensional elements, the natural coordinate axes (x, h) are used which have their origin at the centre of the element (x=0, h=0) and pass through the midpoints of the opposite sides. They need not be orthogonal or parallel to any global coordinate system.

The natural coordinates for such a plane element are given in the following diagram. Note that their orientation is determined by the element node numbering.

Finite Element Theory Contents

Shape Functions

Isoparametric Finite Element Formulation

Numerical Integration

innovative | flexible | trusted

LUSAS is a trademark and trading name of Finite Element Analysis Ltd. Copyright 1982 - 2022. Last modified: November 29, 2022 . Privacy policy. 
Any modelling, design and analysis capabilities described are dependent upon the LUSAS software product, version and option in use.